Advertisements
Advertisements
Question
Prove the following identity :
`(cot^2θ(secθ - 1))/((1 + sinθ)) = sec^2θ((1-sinθ)/(1 + secθ))`
Advertisements
Solution
LHS = `(cot^2θ(secθ - 1))/((1 + sinθ)) `
= `(cot^2θ(secθ - 1)(1 - sinθ)(secθ + 1))/((1 + sinθ)(1 - sinθ)(secθ + 1))`
= `(cot^2θ(secθ - 1)(secθ + 1)(1 - sinθ))/((1 + sinθ)(1 - sinθ)(secθ + 1))`
= `(cot^2θ(sec^2θ - 1)(1 - sinθ))/((1 - sin^2θ)(1 + secθ))`
= `(cot^2θ(tan^2θ)(1 - sinθ))/((cos^2θ)(1 + secθ))` (∵ `tan^2θ = sec^2θ - 1,1 - sin^2θ = cos^2θ`)
= `((cotθtanθ)^2(1 - sinθ))/((cos^2θ)(1 + secθ))`
= `(1(1 - sinθ))/((cos^2θ)(1 + secθ))` (∵ cotθtanθ = 1)
= `sec^2θ((1 - sinθ)/(1 + secθ))`
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities.
`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
Prove the following identities:
(1 + cot A – cosec A)(1 + tan A + sec A) = 2
If `( tan theta + sin theta ) = m and ( tan theta - sin theta ) = n " prove that "(m^2-n^2)^2 = 16 mn .`
Define an identity.
Write True' or False' and justify your answer the following :
The value of \[\sin \theta\] is \[x + \frac{1}{x}\] where 'x' is a positive real number .
Prove that: 2(sin6 θ + cos6 θ) – 3 (sin4 θ + cos4 θ) + 1 = 0.
Prove that tan2Φ + cot2Φ + 2 = sec2Φ.cosec2Φ.
Prove that `"cosec" θ xx sqrt(1 - cos^2theta)` = 1
Prove the following:
`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A
